Separoid

In mathematics, a separoid is a binary relation between disjoint sets which is stable as an ideal in the canonical order induced by inclusion. Many mathematical objects which appear to be quite different, find a common generalisation in the framework of separoids; e.g., graphs, configurations of convex sets, oriented matroids, and polytopes. Any countable category is an induced subcategory of separoids when they are endowed with homomorphisms (viz., mappings that preserve the so-called minimal Radon partitions). In mathematics, a separoid is a binary relation between disjoint sets which is stable as an ideal in the canonical order induced by inclusion. Many mathematical objects which appear to be quite different, find a common generalisation in the framework of separoids; e.g., graphs, configurations of convex sets, oriented matroids, and polytopes. Any countable category is an induced subcategory of separoids when they are endowed with homomorphisms (viz., mappings that preserve the so-called minimal Radon partitions). In this general framework, some results and invariants of different categories turn out to be special cases of the same aspect; e.g., the pseudoachromatic number from graph theory and the Tverberg theorem from combinatorial convexity are simply two faces of the same aspect, namely, complete colouring of separoids. A separoid is a set S {displaystyle S} endowed with a binary relation ∣   ⊆ 2 S × 2 S {displaystyle mid subseteq 2^{S} imes 2^{S}} on its power set, which satisfies the following simple properties for A , B ⊆ S {displaystyle A,Bsubseteq S} : A related pair A ∣ B {displaystyle Amid B} is called a separation and we often say that A is separated from B. It is enough to know the maximal separations to reconstruct the separoid. A mapping φ : S → T {displaystyle varphi colon S o T} is a morphism of separoids if the preimages of separations are separations; that is, for A , B ⊆ T {displaystyle A,Bsubseteq T} Examples of separoids can be found in almost every branch of mathematics. Here we list just a few. 1. Given a graph G=(V,E), we can define a separoid on its vertices by saying that two (disjoint) subsets of V, say A and B, are separated if there are no edges going from one to the other; i.e., 2. Given an oriented matroid M = (E,T), given in terms of its topes T, we can define a separoid on E by saying that two subsets are separated if they are contained in opposite signs of a tope. In other words, the topes of an oriented matroid are the maximal separations of a separoid. This example includes, of course, all directed graphs. 3. Given a family of objects in an Euclidean space, we can define a separoid in it by saying that two subsets are separated if there exists a hyperplane that separates them; i.e., leaving them in the two opposite sides of it.